Properties of exponents- PRODUCT OF POWERS PROPERTY: a^m * a^n = a^m+n POWER OF A POWER PROPERTY: (a^m) = a^mn POWER OF A PRODUCT PROPERTY: (ab)^m = a^m * b^m NEGATIVE EXPONENT PROPERTY: a^-m = 1/a^m, "a" cannot be 0 QUOTIENT OF POWERS PROPERTY: a^m/a^n = a^m-n, "a" cannot be 0 POWER OF A QUOTIENT PROPERTY: (a/b)^m = a^m/b^m, "b" cannot be 0
Scientific Notation- SCIENTIFIC NOTATION: when a number is written in the form c * 10^n where c is greater than or equal to 1 and is less than 10 and n is an integer
For example, the width of a molecule of water is about 2.5 X 10^-8 meter, or 0.000000025 meter. When working with numbers in scientific notation, the properties of exponents can help with calculations.
6.2-Evaluation and Graphing Polynomial Functions
POLYNOMIAL FUNCTION: a function in the form f(x) = ax^4 + bx^3 + cx^2 + dx + e where a cannot be 0, the exponents are all whole numbers, and the coefficients are all real numbers. In this polynomial function a is the LEADING COEFFICIENT, e is the CONSTANT TERM, and 4 is the DEGREE. A polynomial in STANDARD FORM is written so as all of its terms are in descending order of exponents from left to right.
Examples of Degree -
f(x) = 3x + 2 is a polynomial function of degree 1
f(x) = x^2 + 3x +2 is a polynomial function of degree 2
Methods for Solving- DIRECT SUBSTITUTION- substitute the given value of x into the polynomial function in order to solve for f(x) SYNTHETIC SUBSTITUTION- write the given value of x and take note of the coefficients of f(x). Bring down the leading coefficient and multiply it by the given x value then add the result to the next coefficient in order. Take that sum and multiply that by the given x value and add it to the next coefficient, continue this until you are left with the solution.
Example of Synthetic Substitution-
f(x) = 2X^4 - 8X^2 + 5X - 7, when x = 3
Drop down leading coefficients: 2 0 -8 5 -7
Multiply them by x(3) and add: 6 18 30 105
Added totals (solution is last) :2 6 10 35 98
Solution when x = 3: f(3) = 98
Graphing Polynomial Functions- END BEHAVIOR- the behavior of the graph as x approaches positive infinity ( +∞) or negative infinity ( -∞). The expression x--> +∞ is read as "x approaches positive infinity."
End Behavior for Polynomial Functions (based on the degree and leading coefficient)-
Negative leading coefficient and even degree: x--> +∞, f(x)--> -∞ and x--> -∞, f(x)--> -∞
Positive leading coefficient and even degree: x--> +∞, f(x)--> +∞ and x--> -∞, f(x)--> +∞
Negative leading coefficient and odd degree: x--> +∞, f(x)--> -∞ and x--> -∞, f(x)--> +∞
Positive leading coefficient and odd degree: x--> +∞, f(x)--> +∞ and x--> -∞, f(x)--> -∞
To graph:
1. Determine the end behavior of the graph to help check the graph.
2. Pick a few points around the origin to plug into the function and solve for.
3. Graph those points and see if they match your end behavior, if so, good job!
6.3- Adding, Subtracting, and Multiplying Polynomials
Adding To add polynomials, all you do is combine like terms, it is always helpful to align like terms in a verticle column as well.
For example:
5x^4 - 2x^3 + 10x^2 - 7x - 17
+ 6x^3 - 14x^2 + 20
=5x^4 + 4x^3 - 4x^2 - 7x + 3
Subtract To subtract polynomials, you change the subtraction sign in the problem to an addition sign and make the terms following the new addition sign opposite of what they were.
For example:
(5x^2 + 4x^3 - 7) - (4x - 7x^5 + 9x^2) original problem
(5x^2 + 4x^3 - 7) + (-4x + 7x^5 -9x^2) switch signs
7x^5 + 4x^3 - 4x^2 - 4x - 7 add like terms
Multiply To multiply polynomials use foil to multiply everything together.
For example:
(x - 3) (4x^2 - 5) (2x^3 +1) original problem
4x^3 -5x -12x^2 +15 (2x^3 +1) foil any pair of terms
8x^6 + 4x^3 - 10x^4 - 5x - 24x^5 - 12x + 30x^3 +15 foil the terms left
8x^6 - 24x^5 - 10x^4 + 34x^3 -17x + 15 combine like terms
Special Product Patterns Example Sum and Difference- (a + b)^2 (a - b)^2 = a^2 - b^2 (x - 3)(x + 3) = x^2 - 9 Square of a Binomal- (a + b)^2 = a^2 + 2ab + b^2 (y + 4)^2 = y^2 + 8y + 16
(a - b)^2 = a^2 - 2ab + b^2 (3t^2 - 2)^2 = 9t^4 - 12 t^2 + 4 Cube of a Binomial- (a + b)^3 = a^3 + 3a^2b + 3ab^2 =b^3 (x + 1)^3 = x^3 + 3x^2 + 3x + 1
(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 (p - 2)^3 = p^3 - 6p^2 + 12p - 8
6.4- Factoring and Solving Polynomial Equations
Factoring (* are the new ones):
Type
Example
General Trinomial
2x^2 - 5x - 12 = (2x + 3)(x - 4)
Perfect Square Trinomial
x^2 + 10x + 25 = (x + 5)^2
Difference of Two Squares
4x^2 - 9 = (2x + 3)(2x - 3)
Common Monomial Factor
6x^2 + 15x = 3x(2x + 5)
*Sum of Two Cubes
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
*Difference of Two Cubes
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
*Example of Sum of Two Cubes- x^3 + 8 = (x+ 2)(x^2 - 2x + 4)
*Example of Difference of Two Cubes- 8x^3 - 1 = (2x - 1)(4x^2 + 2x + 1)
Solving Polynomials by Factoring-
Solving polynomials in this section is just about identical to the way you are used to, once the equation is fully factored, set all terms equal to 0 and solve, this will give you the x solutions.
Example- 3x^3 - 12x^2 + 2x - 8 = 0
3x^2(x - 4) + 2(x - 4) = 0
(3x^2 + 2)(x - 4) = 0
3x^2 + 2 = 0, x = i√2/3 and -i√2/3
x - 4 = 0, x = 4
x = 4, i √2/3, -i √2/3
6.5 The Remainder and Factor Theorems
The Remainder and Factor Theorems
When you divide a polynomial f(x) by a divisor d(x), you get a quotient polynomial q(x) and the remainder polynimial r(x). We write this as
f(x)/d(x) = q(x) + r(x)/d(x). The degree of the remainder must be less than the degree of the divisor. Remainder Theorem- if a polynomial f(x) is divided by x - k, then the remainder is r = f(k).
Polynomial Long Division-
Divide 2x^4 + 3x^3 + 5x - 1 by x^2 - 2x + 2
Always use first terms, no other part of the polynomials(terms with the highest powers)
Figure out what times the first term of the divisor willl get it closest to the first term of the divisor and write that up top.
Then multiply all the terms in the divisor by that number and subtract that from the dividend.
Continue to do this until you are left with a remainder polynomial with a degree that is less than the divisor.
Write the remainder polynomial over the divisor
The solution can be checked by multiplying the divisor by the quotient and adding the remainder, the result should be the dividend.
Synthetic Division-
Synthetic division is much like synthetic substitution. When you have a binomial divisor, you just find the zero of that expression and use that number to multiply the others by in your synthetic division. (The zero of the expression x - 2 is 2, because if x = 2, then the whole expression will equal 0). The bottom row of numbers that result from your synthetic division are the coefficients of the polynomial quotient adn the last number on the right is the remainder.
Example-
x^4 - 2x^3 + 5x + 2 = 0 divided by x - 4, the zero would be 4
4] ------>1 -2 0 5 2
------------->4 8 32 148
---------> 1 2 8 37 150
Therefore the quotient is:
x^3 + 2x^2 + 8x + 37 + 150/x-4
*150/x-4 is the remainder
6.6 Finding Rational Zeros
Rational Zero Theorem- If f(x) = ax^n + ...... bx + c has integer coefficients, then every rational xero of f has the following form:
p/q = factors of constant term (c) / factors of leading coefficient (a^n)
Example:
Fing the rational zeros of f(x) = x^3 + 2x^2 - 11x - 12
List the possible rational zeros, The leading coefficient is a 1 and the constant is -12. The possible rational zeros are:
x = +/- (1, 2, 3, 4, 6, 12)
Test these possible zeros by synthetic division:
1| 1 2 -11 -12 | 1 3 -8 1 3 -8 -20 There is a remainder, therefore 1 is not a zero. -1| 1 2 -11 -12 | -1 -1 12 1 1 -12 0 There is no remainder, therefore -1 is a zero.
Since -1 is a zero of f, you can write the following:
f(x) = (x + 1)(x^2 + x -12)
This can be factored:
f(x) = (x + 1)(x - 3)(x + 4)
Therefore the zeros of f are -1, 3, and -4.
*HINT- The original polynomial equation can be plugged into the y= screen of a graphing calculator to find the zeros as well.
6.7 Using the Fundamental Theorem of Algebra
The Fundamental Theorem of Algebra- If f(x) is a polynomial of degree n where n > 0, then the equation f(x) = 0 has at least one root in the set of complex mumbers. (A polynomial will have the same number of solutions as the degree of the polynomial = x^3 + 7 = f(x) has 3 solutions)
Repeated Solution- a solution of a polynomial that repeats, but is still counted towards the total number of solutions.
This polynomial has 5 solutions and is of the 5th degree.
*Real zeros will cross the x-axis, however imaginary solutions will not cross the x-axis.
6.8 Analyzing Graphs of Polynomial Functions
Summary of Concepts:
Zero- a number k is a zero if f(x) = 0 Factor- x - k is a factor of f(x) Solution- k is a solution of the polynomial equation f(x) = 0. If k is a real number, then the following is also equivalent. X-intercept- k is an x-intercept of the graph of the polynomial function f (k is a zero)
The key concepts learned in previous chapters can be used to graph polynomial functions.
Factoring can be used to find the x-intercepts of the polynomial's graph
After finding those, you can determine the end behavior of the polynomial to be checked with the graph later
Plot those intercepts and if needed use a table to find other points on the graph
Check the end behavior with your final graph
Practice Problems 6.1 Using properties of Exponents Example: (8^2)^3
Solve: (8^2)^3 = 8^2+3 = 8^5 = 8×8×8×8×8 = 32768
Now you try:
17. (5^-2)^3
21. (3/7)^3
Answers: The Back of Your Book p SA 20 6.2 Evaluating and Graphing Polynomial Functions
Example
Direct Substitution -
f(x)=2x^4 - 8x^2 + 5x - 7, where x=2
=2(2)^4 - 8(2)^2 + 5(2) - 7
=32 - 32 + 10 - 7
=3
Now you try:
33. 11x^3 - 6x^2 + 2 where x=0
35. 7x^3 + 9x^2 + 3x where x=10
Answers: The Back of Your Book, p SA 21 6.3 Adding, Subtracting and Multiplying Polynomials
Adding Example -
(3x^3 + 2x^2 - x - 7) + (x^3 - 10x^2 + 8) = 4x^3 + 8x^2 - x + 1
Now you try:
19. (4x^2 - 11x + 10) + (5x - 31)
25. (10x - 3 + 7x^2) + (x^3 - 2x +17)
Answers: The Back of Your Book p. SA 21
Subtracting Example -
(8x^3 - 3x^2 - 2x +9) - (2x^3 + 6x^2 - x + 1) = (8x^3 - 3x^2 - 2x +9) + (-2x^3 - 6x^2 + x - 1) = (6x^3 - 9x^2 - x
+8)
Now you try:
23. (10x^3 - 4x^2 + 3x) - (x^3 - 2x + 17)
15. (x^2 - 6x + 5) - (x^2 + x - 2)
Answers: The Back of Your Book p. SA 21
Multiplying Example:
(x - 3)(3x^2 - 2x - 4) = 3x^3 - 11x^2 + 2x + 12
Now you try:
41. (3x^2 - 2)(x^2 + 4x + 3)
39. (x - 1)(x^3 + 2x^2 + 2)
Answers: The Back of Your Book p.SA 21
6.4 Factoring and Solving Polynomial Equations Example:
2x^5 + 24x = 14x^3
2x^5 + 24x - 14x^3 = 0 Bold - the bolded is the solution to this if it were a factoring problem
2x(x^4 - 7x^2 + 12) = 0
2x(x^2 - 3)(x^2 - 4) = 0 2x(x^2 - 3)(x - 2)(x + 2) = 0
x = 0, ± square root of 3, -2, 2
Now you try:
79. x^3 - 15x^2 + 5x - 25 = 0
61. 4x^4 + 39x^2 - 10
6.5 The Remainder and Factor Theorems Example:
(x^3 + 2x^2 - 6x - 9) / (x -2)
Now you try:
47. f(x) = 2x^3 + 4x^2 - 2x - 4
53. f(x) = 2x^4 + 3x^3 - 3x^2 + 3x - 5
Answers: The Back of Your Book p SA 22 6.7 Using the Fundamental Theorem of Algebra Example: f(x) = x^5 - 2^4 + 8x^2 - 13x + 6
all possible zeros: ±1, ±2, ±3, ±6 *after synthetic division (to show the theorem rather than repeat 6.6)*
f(x) = (x - 1)(x - 1)(x + 2)(x^2 - 2x + 3)
f(x) = (x - 1)(x - 1)(x + 2)[x - (1 + i|¯2)][x - (1 - i |¯2)]
x = 1, 1, -2, 1 ± i|¯2
Now you try:
29. x^4 + 6x^3 +14x^2 +54x + 45
31. x^4 - x^3 - 5x^2 - x - 6
Answers: The Back of Your Book p SA 23
6.8 Analyzing Graphs of Polynomial Functions Example: x | y
-4 | -12 (3/4)
-3 | -4
-1 | 1
0 | (1/2)
2 | 1
3 | 5 *insert graph here* (due to technical difficulty, the graph before you is nonexistent)
Now you try:
graph the functions
17. f(x) = 5(x - 1)(x - 2)(x - 3)
21 f(x) = (x - 2)(x^2 + x + 1)
Answers: The Back of Your Book p SA 23 Careers That Use Polynomials Wildlife biologists use a lot of polynomials with their job, such as Ornithologists, who study birds. They collect large amounts of data and information that they use to describe bird populations and characteristics, and they create mathematical models that help determine the size and growth of bird populations. People in the medical field, like nurses, also use math on their job when dealing with medication,taking vital signs, making assessments, and helping to make treatment plans. Gerontologists use math in their jobs when providing financial advice, creating budgets, or when studying demographic data and interpreting statistics. Archeologists develop and test hypotheses based on material remains, and statistics are used when they compile and analyze large amounts of data gathered during their fieldwork. There are many careers that use polynomials with their job.
Chapter 6- Polynomials and Polynomial Functions
6.1-Using Properties of Exponents
Properties of exponents-
PRODUCT OF POWERS PROPERTY: a^m * a^n = a^m+n
POWER OF A POWER PROPERTY: (a^m) = a^mn
POWER OF A PRODUCT PROPERTY: (ab)^m = a^m * b^m
NEGATIVE EXPONENT PROPERTY: a^-m = 1/a^m, "a" cannot be 0
QUOTIENT OF POWERS PROPERTY: a^m/a^n = a^m-n, "a" cannot be 0
POWER OF A QUOTIENT PROPERTY: (a/b)^m = a^m/b^m, "b" cannot be 0
Scientific Notation-
SCIENTIFIC NOTATION: when a number is written in the form c * 10^n where c is greater than or equal to 1 and is less than 10 and n is an integer
For example, the width of a molecule of water is about 2.5 X 10^-8 meter, or 0.000000025 meter. When working with numbers in scientific notation, the properties of exponents can help with calculations.
6.2-Evaluation and Graphing Polynomial Functions
POLYNOMIAL FUNCTION: a function in the form f(x) = ax^4 + bx^3 + cx^2 + dx + e where a cannot be 0, the exponents are all whole numbers, and the coefficients are all real numbers. In this polynomial function a is the LEADING COEFFICIENT, e is the CONSTANT TERM, and 4 is the DEGREE. A polynomial in STANDARD FORM is written so as all of its terms are in descending order of exponents from left to right.
Examples of Degree -
f(x) = 3x + 2 is a polynomial function of degree 1
f(x) = x^2 + 3x +2 is a polynomial function of degree 2
Common Types of Polynomial Functions-
http://www.teachertube.com/view_video.php?viewkey=e258cee61f4377e5bf3e
Methods for Solving-
DIRECT SUBSTITUTION- substitute the given value of x into the polynomial function in order to solve for f(x)
SYNTHETIC SUBSTITUTION- write the given value of x and take note of the coefficients of f(x). Bring down the leading coefficient and multiply it by the given x value then add the result to the next coefficient in order. Take that sum and multiply that by the given x value and add it to the next coefficient, continue this until you are left with the solution.
Example of Synthetic Substitution-
f(x) = 2X^4 - 8X^2 + 5X - 7, when x = 3
Drop down leading coefficients: 2 0 -8 5 -7
Multiply them by x(3) and add: 6 18 30 105
Added totals (solution is last) :2 6 10 35 98
Solution when x = 3: f(3) = 98
Graphing Polynomial Functions-
END BEHAVIOR- the behavior of the graph as x approaches positive infinity ( +∞) or negative infinity ( -∞). The expression x--> +∞ is read as "x approaches positive infinity."
End Behavior for Polynomial Functions (based on the degree and leading coefficient)-
Negative leading coefficient and even degree: x--> +∞, f(x)--> -∞ and x--> -∞, f(x)--> -∞
Positive leading coefficient and even degree: x--> +∞, f(x)--> +∞ and x--> -∞, f(x)--> +∞
Negative leading coefficient and odd degree: x--> +∞, f(x)--> -∞ and x--> -∞, f(x)--> +∞
Positive leading coefficient and odd degree: x--> +∞, f(x)--> +∞ and x--> -∞, f(x)--> -∞
To graph:
1. Determine the end behavior of the graph to help check the graph.
2. Pick a few points around the origin to plug into the function and solve for.
3. Graph those points and see if they match your end behavior, if so, good job!
College Algebra: Graphing Polynomial Functions - The funniest videos are a click away
http://www.metacafe.com/watch/2110596/college_algebra_graphing_polynomial_functions/
6.3- Adding, Subtracting, and Multiplying Polynomials
Adding
To add polynomials, all you do is combine like terms, it is always helpful to align like terms in a verticle column as well.
For example:
5x^4 - 2x^3 + 10x^2 - 7x - 17
+ 6x^3 - 14x^2 + 20
=5x^4 + 4x^3 - 4x^2 - 7x + 3
Subtract
To subtract polynomials, you change the subtraction sign in the problem to an addition sign and make the terms following the new addition sign opposite of what they were.
For example:
(5x^2 + 4x^3 - 7) - (4x - 7x^5 + 9x^2) original problem
(5x^2 + 4x^3 - 7) + (-4x + 7x^5 -9x^2) switch signs
7x^5 + 4x^3 - 4x^2 - 4x - 7 add like terms
Multiply
To multiply polynomials use foil to multiply everything together.
For example:
(x - 3) (4x^2 - 5) (2x^3 +1) original problem
4x^3 -5x -12x^2 +15 (2x^3 +1) foil any pair of terms
8x^6 + 4x^3 - 10x^4 - 5x - 24x^5 - 12x + 30x^3 +15 foil the terms left
8x^6 - 24x^5 - 10x^4 + 34x^3 -17x + 15 combine like terms
Special Product Patterns
Example
Sum and Difference- (a + b)^2 (a - b)^2 = a^2 - b^2 (x - 3)(x + 3) = x^2 - 9
Square of a Binomal- (a + b)^2 = a^2 + 2ab + b^2 (y + 4)^2 = y^2 + 8y + 16
(a - b)^2 = a^2 - 2ab + b^2 (3t^2 - 2)^2 = 9t^4 - 12 t^2 + 4
Cube of a Binomial- (a + b)^3 = a^3 + 3a^2b + 3ab^2 =b^3 (x + 1)^3 = x^3 + 3x^2 + 3x + 1
(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 (p - 2)^3 = p^3 - 6p^2 + 12p - 8
6.4- Factoring and Solving Polynomial Equations
Factoring (* are the new ones):
*Example of Difference of Two Cubes- 8x^3 - 1 = (2x - 1)(4x^2 + 2x + 1)
Solving Polynomials by Factoring-
Solving polynomials in this section is just about identical to the way you are used to, once the equation is fully factored, set all terms equal to 0 and solve, this will give you the x solutions.
Example- 3x^3 - 12x^2 + 2x - 8 = 0
3x^2(x - 4) + 2(x - 4) = 0
(3x^2 + 2)(x - 4) = 0
3x^2 + 2 = 0, x = i√2/3 and -i√2/3
x - 4 = 0, x = 4
x = 4, i √2/3, -i √2/3
6.5 The Remainder and Factor Theorems
The Remainder and Factor Theorems
When you divide a polynomial f(x) by a divisor d(x), you get a quotient polynomial q(x) and the remainder polynimial r(x). We write this as
f(x)/d(x) = q(x) + r(x)/d(x). The degree of the remainder must be less than the degree of the divisor.
Remainder Theorem- if a polynomial f(x) is divided by x - k, then the remainder is r = f(k).
Polynomial Long Division-
Divide 2x^4 + 3x^3 + 5x - 1 by x^2 - 2x + 2
--------------------------------------> 2x^2 + 7x + 10
------------->
x^2 - 2x + 2 | 2x^4 + 3x^3 + 0x^2 + 5x - 1
---------------->+ -2x^4 + 4x^3 - 2x^2
----------------> = 7x^3 - 2x^2 + 5x - 1
--------------------------->+ -7x^3 + 14x^2 -14x + 0
--------------------------->= 10x^2 - 9x - 1
--------------------------------------->+ -10x^2 + 20x - 20
--------------------------------------->= 11x - 21
2x^4 + 3x^3 + 5x - 1/ x^2 - 2x + 2 = 2x^2 + 7x + 10 + 11x -21/x^2 - 2x + 2
Always use first terms, no other part of the polynomials(terms with the highest powers)
Figure out what times the first term of the divisor willl get it closest to the first term of the divisor and write that up top.
Then multiply all the terms in the divisor by that number and subtract that from the dividend.
Continue to do this until you are left with a remainder polynomial with a degree that is less than the divisor.
Write the remainder polynomial over the divisor
The solution can be checked by multiplying the divisor by the quotient and adding the remainder, the result should be the dividend.
Synthetic Division-
Synthetic division is much like synthetic substitution. When you have a binomial divisor, you just find the zero of that expression and use that number to multiply the others by in your synthetic division. (The zero of the expression x - 2 is 2, because if x = 2, then the whole expression will equal 0). The bottom row of numbers that result from your synthetic division are the coefficients of the polynomial quotient adn the last number on the right is the remainder.
Example-
x^4 - 2x^3 + 5x + 2 = 0 divided by x - 4, the zero would be 4
4] ------>1 -2 0 5 2
------------->4 8 32 148
---------> 1 2 8 37 150
Therefore the quotient is:
x^3 + 2x^2 + 8x + 37 + 150/x-4
*150/x-4 is the remainder
6.6 Finding Rational Zeros
Rational Zero Theorem- If f(x) = ax^n + ...... bx + c has integer coefficients, then every rational xero of f has the following form:
p/q = factors of constant term (c) / factors of leading coefficient (a^n)
Example:
Fing the rational zeros of f(x) = x^3 + 2x^2 - 11x - 12
List the possible rational zeros, The leading coefficient is a 1 and the constant is -12. The possible rational zeros are:
x = +/- (1, 2, 3, 4, 6, 12)
Test these possible zeros by synthetic division:
1| 1 2 -11 -12
| 1 3 -8
1 3 -8 -20
There is a remainder, therefore 1 is not a zero.
-1| 1 2 -11 -12
| -1 -1 12
1 1 -12 0
There is no remainder, therefore -1 is a zero.
Since -1 is a zero of f, you can write the following:
f(x) = (x + 1)(x^2 + x -12)
This can be factored:
f(x) = (x + 1)(x - 3)(x + 4)
Therefore the zeros of f are -1, 3, and -4.
*HINT- The original polynomial equation can be plugged into the y= screen of a graphing calculator to find the zeros as well.
6.7 Using the Fundamental Theorem of Algebra
The Fundamental Theorem of Algebra- If f(x) is a polynomial of degree n where n > 0, then the equation f(x) = 0 has at least one root in the set of complex mumbers. (A polynomial will have the same number of solutions as the degree of the polynomial = x^3 + 7 = f(x) has 3 solutions)
Repeated Solution- a solution of a polynomial that repeats, but is still counted towards the total number of solutions.
Example- Find all of the zeros of:
f(x) = x^5 - 2x^4 + 8x^2 - 13x + 6
f(x) = (x - 1)(x - 1)(x + 2)(x^2 - 2x + 3)
f(x) = (x - 1)(x - 1)(x + 2)[x - (1 + i√2)][x - (1 - i√2)]
x = 1, 1, -2, 1 + i√2, 1 - i√2
This polynomial has 5 solutions and is of the 5th degree.
*Real zeros will cross the x-axis, however imaginary solutions will not cross the x-axis.
6.8 Analyzing Graphs of Polynomial Functions
Summary of Concepts:
Zero- a number k is a zero if f(x) = 0
Factor- x - k is a factor of f(x)
Solution- k is a solution of the polynomial equation f(x) = 0. If k is a real number, then the following is also equivalent.
X-intercept- k is an x-intercept of the graph of the polynomial function f (k is a zero)
The key concepts learned in previous chapters can be used to graph polynomial functions.
Practice Problems
6.1 Using properties of Exponents
Example: (8^2)^3
Solve: (8^2)^3 = 8^2+3 = 8^5 = 8×8×8×8×8 = 32768
Now you try:
17. (5^-2)^3
21. (3/7)^3
Answers: The Back of Your Book p SA 20
6.2 Evaluating and Graphing Polynomial Functions
Example
Direct Substitution -
f(x)=2x^4 - 8x^2 + 5x - 7, where x=2
=2(2)^4 - 8(2)^2 + 5(2) - 7
=32 - 32 + 10 - 7
=3
Now you try:
33. 11x^3 - 6x^2 + 2 where x=0
35. 7x^3 + 9x^2 + 3x where x=10
Answers: The Back of Your Book, p SA 21
6.3 Adding, Subtracting and Multiplying Polynomials
Adding Example -
(3x^3 + 2x^2 - x - 7) + (x^3 - 10x^2 + 8) = 4x^3 + 8x^2 - x + 1
Now you try:
19. (4x^2 - 11x + 10) + (5x - 31)
25. (10x - 3 + 7x^2) + (x^3 - 2x +17)
Answers: The Back of Your Book p. SA 21
Subtracting Example -
(8x^3 - 3x^2 - 2x +9) - (2x^3 + 6x^2 - x + 1) = (8x^3 - 3x^2 - 2x +9) + (-2x^3 - 6x^2 + x - 1) = (6x^3 - 9x^2 - x
+8)
Now you try:
23. (10x^3 - 4x^2 + 3x) - (x^3 - 2x + 17)
15. (x^2 - 6x + 5) - (x^2 + x - 2)
Answers: The Back of Your Book p. SA 21
Multiplying Example:
(x - 3)(3x^2 - 2x - 4) = 3x^3 - 11x^2 + 2x + 12
Now you try:
41. (3x^2 - 2)(x^2 + 4x + 3)
39. (x - 1)(x^3 + 2x^2 + 2)
Answers: The Back of Your Book p.SA 21
6.4 Factoring and Solving Polynomial Equations
Example:
2x^5 + 24x = 14x^3
2x^5 + 24x - 14x^3 = 0 Bold - the bolded is the solution to this if it were a factoring problem
2x(x^4 - 7x^2 + 12) = 0
2x(x^2 - 3)(x^2 - 4) = 0
2x(x^2 - 3)(x - 2)(x + 2) = 0
x = 0, ± square root of 3, -2, 2
Now you try:
79. x^3 - 15x^2 + 5x - 25 = 0
61. 4x^4 + 39x^2 - 10
6.5 The Remainder and Factor Theorems
Example:
(x^3 + 2x^2 - 6x - 9) / (x -2)
2| 1 2 -6 -9
2 8 4
¯¯¯¯¯¯¯¯¯¯¯
1 4 2 -5
x^2 + 4x + 2 + (-5/x-2)
Now you try:
31. (2x^2 + 7x + 8) / (x - 2)
35. (10x^4 + 5x^3 + 4x^2 - 9) / (x + 1)
Answers: The Back of Your Book p SA 22
6.6 Finding Rational Zeros
Example: f(x) = x^3 + 2x^2 - 11x - 12
all possible zeros: ±1, ±2, ±3, ±4, ±6, ±12
-1| 1 2 -11 -12
-1 -1 12
1 1 -12 0
f(x) = (x + 1)(x^2 + x - 12)
f(x) = (x + 1)(x - 3)(x + 4)
x = -1, 3, -4
Now you try:
47. f(x) = 2x^3 + 4x^2 - 2x - 4
53. f(x) = 2x^4 + 3x^3 - 3x^2 + 3x - 5
Answers: The Back of Your Book p SA 22
6.7 Using the Fundamental Theorem of Algebra
Example: f(x) = x^5 - 2^4 + 8x^2 - 13x + 6
all possible zeros: ±1, ±2, ±3, ±6
*after synthetic division (to show the theorem rather than repeat 6.6)*
f(x) = (x - 1)(x - 1)(x + 2)(x^2 - 2x + 3)
f(x) = (x - 1)(x - 1)(x + 2)[x - (1 + i|¯2)][x - (1 - i |¯2)]
x = 1, 1, -2, 1 ± i|¯2
Now you try:
29. x^4 + 6x^3 +14x^2 +54x + 45
31. x^4 - x^3 - 5x^2 - x - 6
Answers: The Back of Your Book p SA 23
6.8 Analyzing Graphs of Polynomial Functions
Example:
x | y
-4 | -12 (3/4)
-3 | -4
-1 | 1
0 | (1/2)
2 | 1
3 | 5
*insert graph here* (due to technical difficulty, the graph before you is nonexistent)
Now you try:
graph the functions
17. f(x) = 5(x - 1)(x - 2)(x - 3)
21 f(x) = (x - 2)(x^2 + x + 1)
Answers: The Back of Your Book p SA 23
Careers That Use Polynomials
Wildlife biologists use a lot of polynomials with their job, such as Ornithologists, who study birds. They collect large amounts of data and information that they use to describe bird populations and characteristics, and they create mathematical models that help determine the size and growth of bird populations. People in the medical field, like nurses, also use math on their job when dealing with medication,taking vital signs, making assessments, and helping to make treatment plans. Gerontologists use math in their jobs when providing financial advice, creating budgets, or when studying demographic data and interpreting statistics. Archeologists develop and test hypotheses based on material remains, and statistics are used when they compile and analyze large amounts of data gathered during their fieldwork. There are many careers that use polynomials with their job.